A355734
Least k such that there are exactly n multisets that can be obtained by choosing a divisor of each prime index of k.
Original entry on oeis.org
1, 3, 7, 13, 21, 35, 39, 89, 133, 105, 91, 195, 351, 285, 247, 333, 273, 481, 455, 555, 623, 801, 791, 741, 1359, 1157, 1281, 1335, 1365, 1443, 1977, 1729, 1967, 1869, 2109, 3185, 2373, 2769, 2639, 4361, 3367, 3653, 3885, 3471, 4613, 5883, 5187, 5551, 6327
Offset: 1
The terms together with their prime indices begin:
1: {}
3: {2}
7: {4}
13: {6}
21: {2,4}
35: {3,4}
39: {2,6}
89: {24}
133: {4,8}
105: {2,3,4}
91: {4,6}
195: {2,3,6}
351: {2,2,2,6}
For example, the choices for a(12) = 195 are:
{1,1,1} {1,2,2} {1,3,6}
{1,1,2} {1,2,3} {2,2,3}
{1,1,3} {1,2,6} {2,3,3}
{1,1,6} {1,3,3} {2,3,6}
Positions of first appearances in
A355733.
A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]],{n,1000}];
Table[Position[az,k][[1,1]],{k,mnrm[az]}]
A370595
Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 3, 2, 4, 3, 4, 5, 8, 9, 8, 13, 12, 17, 16, 27, 28, 33, 36, 39, 50, 58, 65, 75, 93, 94, 112, 125, 148, 170, 190, 209, 250, 273, 305, 341, 403, 432, 484, 561, 623, 708, 765, 873, 977, 1109, 1178, 1367, 1493, 1669, 1824, 2054, 2265, 2521, 2770
Offset: 0
The a(1) = 1 through a(15) = 13 partitions (A = 10, B = 11, C = 12, D = 13):
1 . 21 22 . 33 322 71 441 55 533 B1 553 77 933
31 51 421 332 522 442 722 444 733 D1 B22
321 422 531 721 731 552 751 B21 B31
521 4321 4322 4332 931 4433 4443
5321 4431 4432 5441 5442
5322 5332 6332 5532
5421 5422 7322 6621
6321 6322 7421 7332
7321 7422
7521
8421
9321
54321
The version for prime factors (not all divisors) is
A370594, ranks
A370647.
These partitions have ranks
A370810.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A370592 counts partitions with choosable prime factors, ranks
A368100.
A370593 counts partitions without choosable prime factors, ranks
A355529.
A370804 counts non-condensed partitions with no ones, complement
A370805.
A370814 counts factorizations with choosable divisors, complement
A370813.
-
Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,0,30}]
A371165
Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).
Original entry on oeis.org
3, 5, 11, 17, 26, 31, 35, 38, 39, 41, 49, 57, 58, 59, 65, 67, 69, 77, 83, 86, 87, 94, 109, 119, 127, 129, 133, 146, 148, 157, 158, 179, 191, 202, 206, 211, 217, 235, 237, 241, 244, 253, 274, 277, 278, 283, 284, 287, 291, 298, 303, 319, 326, 331, 333, 334, 353
Offset: 1
The terms together with their prime indices begin:
3: {2} 67: {19} 158: {1,22}
5: {3} 69: {2,9} 179: {41}
11: {5} 77: {4,5} 191: {43}
17: {7} 83: {23} 202: {1,26}
26: {1,6} 86: {1,14} 206: {1,27}
31: {11} 87: {2,10} 211: {47}
35: {3,4} 94: {1,15} 217: {4,11}
38: {1,8} 109: {29} 235: {3,15}
39: {2,6} 119: {4,7} 237: {2,22}
41: {13} 127: {31} 241: {53}
49: {4,4} 129: {2,14} 244: {1,1,18}
57: {2,8} 133: {4,8} 253: {5,9}
58: {1,10} 146: {1,21} 274: {1,33}
59: {17} 148: {1,1,12} 277: {59}
65: {3,6} 157: {37} 278: {1,34}
For prime factors instead of divisors on both sides we get
A319899.
For prime factors on LHS we get
A370802, for distinct prime factors
A371177.
For (greater than) instead of (equal) we get
A371166.
For (less than) instead of (equal) we get
A371167.
Partitions of this type are counted by
A371172.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Select[Range[100],Length[Divisors[#]] == Length[Union@@Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
A371168
Positive integers with fewer prime factors (A001222) than distinct divisors of prime indices (A370820).
Original entry on oeis.org
3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 65, 67, 69, 70, 71, 73, 74, 76, 77, 78, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 111, 113, 114, 115
Offset: 1
The prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
3: {2} 35: {3,4} 59: {17} 86: {1,14}
5: {3} 37: {12} 61: {18} 87: {2,10}
7: {4} 38: {1,8} 65: {3,6} 89: {24}
11: {5} 39: {2,6} 67: {19} 91: {4,6}
13: {6} 41: {13} 69: {2,9} 93: {2,11}
14: {1,4} 43: {14} 70: {1,3,4} 94: {1,15}
15: {2,3} 46: {1,9} 71: {20} 95: {3,8}
17: {7} 47: {15} 73: {21} 97: {25}
19: {8} 49: {4,4} 74: {1,12} 101: {26}
21: {2,4} 51: {2,7} 76: {1,1,8} 103: {27}
23: {9} 52: {1,1,6} 77: {4,5} 105: {2,3,4}
26: {1,6} 53: {16} 78: {1,2,6} 106: {1,16}
29: {10} 55: {3,5} 79: {22} 107: {28}
31: {11} 57: {2,8} 83: {23} 109: {29}
33: {2,5} 58: {1,10} 85: {3,7} 111: {2,12}
For divisors instead of prime factors on the LHS we get
A371166.
The complement is counted by
A371169.
Partitions of this type are counted by
A371173.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A371173
Number of integer partitions of n with fewer parts than distinct divisors of parts.
Original entry on oeis.org
0, 0, 1, 1, 1, 3, 2, 4, 6, 7, 11, 11, 17, 20, 26, 34, 44, 56, 67, 84, 102, 131, 156, 195, 232, 283, 346, 411, 506, 598, 721, 855, 1025, 1204, 1448, 1689, 2018, 2363, 2796, 3265, 3840, 4489, 5242, 6104, 7106, 8280, 9595, 11143, 12862, 14926, 17197, 19862, 22841
Offset: 0
The partition (4,3,2) has 3 parts {2,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
The a(2) = 1 through a(10) = 11 partitions:
(2) (3) (4) (5) (6) (7) (8) (9) (10)
(3,2) (4,2) (4,3) (4,4) (5,4) (6,4)
(4,1) (5,2) (5,3) (6,3) (7,3)
(6,1) (6,2) (7,2) (8,2)
(4,3,1) (8,1) (9,1)
(6,1,1) (4,3,2) (4,3,3)
(6,2,1) (5,3,2)
(5,4,1)
(6,2,2)
(6,3,1)
(8,1,1)
For submultisets instead of parts on the LHS we get ranks
A371166.
These partitions are ranked by
A371168.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Table[Length[Select[IntegerPartitions[n],Length[#] < Length[Union@@Divisors/@#]&]],{n,0,30}]
A370810
Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.
Original entry on oeis.org
1, 2, 6, 9, 10, 22, 25, 30, 34, 42, 45, 62, 63, 66, 75, 82, 98, 99, 102, 110, 118, 121, 134, 147, 153, 166, 170, 186, 210, 218, 230, 246, 254, 275, 279, 289, 310, 314, 315, 330, 343, 354, 358, 363, 369, 374, 382, 390, 402, 410, 422, 425, 462, 482, 490, 495
Offset: 1
The prime indices of 6591 are {2,6,6,6}, for which the only choice is {1,2,3,6}, so 6591 is in the sequence.
The terms together with their prime indices begin:
1: {}
2: {1}
6: {1,2}
9: {2,2}
10: {1,3}
22: {1,5}
25: {3,3}
30: {1,2,3}
34: {1,7}
42: {1,2,4}
45: {2,2,3}
62: {1,11}
63: {2,2,4}
66: {1,2,5}
75: {2,3,3}
82: {1,13}
98: {1,4,4}
99: {2,2,5}
102: {1,2,7}
110: {1,3,5}
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A370814 counts factorizations with choosable divisors, complement
A370813.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Union[Sort /@ Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]]]==1&]
A371172
Number of integer partitions of n with as many submultisets as distinct divisors of parts.
Original entry on oeis.org
0, 0, 1, 1, 0, 1, 0, 3, 2, 3, 1, 4, 2, 1, 2, 3, 4, 2, 4, 1, 5, 2, 7, 5, 9, 4, 9, 15, 18, 16, 24, 13, 17, 23, 23, 22, 34, 17, 30, 31, 36, 29, 43, 21, 30, 35, 44, 28, 47, 19, 44
Offset: 0
The partition (8,6,6) has 6 submultisets {(8,6,6),(8,6),(6,6),(8),(6),()} and 6 distinct divisors of parts {1,2,3,4,6,8}, so is counted under a(20).
The a(17) = 2 through a(24) = 9 partitions:
(17) (9,9) (19) (11,9) (14,7) (13,9) (23) (21,3)
(13,4) (15,3) (15,5) (17,4) (21,1) (19,4) (22,2)
(6,6,6) (8,6,6) (8,8,6) (22,1) (8,8,8)
(12,3,3) (12,4,4) (10,6,6) (15,4,4) (10,8,6)
(18,1,1) (16,3,3) (12,10,1) (12,6,6)
(18,2,2) (12,7,5)
(20,1,1) (18,3,3)
(20,2,2)
(12,10,2)
These partitions are ranked by
A371165.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Table[Length[Select[IntegerPartitions[n], Length[Divisors[Times@@Prime/@#]] == Length[Union@@Divisors/@#]&]],{n,0,30}]
A371127
Powers of 2 times powers > 1 of a prime-indexed prime number.
Original entry on oeis.org
3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 22, 24, 25, 27, 31, 34, 36, 40, 41, 44, 48, 50, 54, 59, 62, 67, 68, 72, 80, 81, 82, 83, 88, 96, 100, 108, 109, 118, 121, 124, 125, 127, 134, 136, 144, 157, 160, 162, 164, 166, 176, 179, 191, 192, 200, 211, 216, 218, 236, 241
Offset: 1
The terms together with their prime indices begin:
3: {2}
5: {3}
6: {1,2}
9: {2,2}
10: {1,3}
11: {5}
12: {1,1,2}
17: {7}
18: {1,2,2}
20: {1,1,3}
22: {1,5}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
31: {11}
34: {1,7}
36: {1,1,2,2}
Subset of
A336101 = powers of 2 times powers of primes.
Counting prime factors instead of divisors gives
A371287.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A076610 lists products of primes of prime index.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A355741 counts choices of a prime factor of each prime index.
A370804
Number of non-condensed integer partitions of n into parts > 1.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 3, 6, 6, 12, 14, 21, 25, 37, 43, 62, 75, 101, 124, 167, 198, 261, 316, 401, 488, 618, 745, 930, 1119, 1379, 1664, 2032, 2433, 2960, 3537, 4259, 5076, 6094, 7227, 8629, 10205, 12126, 14302, 16932, 19893, 23471, 27502, 32315, 37775
Offset: 0
The a(6) = 1 through a(14) = 12 partitions:
(222) . (2222) (333) (3322) (3332) (3333) (4333) (4442)
(3222) (4222) (5222) (4422) (7222) (5333)
(22222) (32222) (6222) (33322) (5522)
(33222) (43222) (8222)
(42222) (52222) (33332)
(222222) (322222) (43322)
(44222)
(53222)
(62222)
(332222)
(422222)
(2222222)
These partitions have as ranks the odd terms of
A355740.
The complement without ones is
A370805.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1] && Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]==0&]],{n,0,30}]
A371170
Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).
Original entry on oeis.org
1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92
Offset: 1
The prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
1: {} 22: {1,5} 42: {1,2,4} 63: {2,2,4}
2: {1} 23: {9} 43: {14} 65: {3,6}
3: {2} 25: {3,3} 45: {2,2,3} 66: {1,2,5}
5: {3} 26: {1,6} 46: {1,9} 67: {19}
6: {1,2} 28: {1,1,4} 47: {15} 69: {2,9}
7: {4} 29: {10} 49: {4,4} 70: {1,3,4}
9: {2,2} 30: {1,2,3} 51: {2,7} 71: {20}
10: {1,3} 31: {11} 52: {1,1,6} 73: {21}
11: {5} 33: {2,5} 53: {16} 74: {1,12}
13: {6} 34: {1,7} 55: {3,5} 75: {2,3,3}
14: {1,4} 35: {3,4} 57: {2,8} 76: {1,1,8}
15: {2,3} 37: {12} 58: {1,10} 77: {4,5}
17: {7} 38: {1,8} 59: {17} 78: {1,2,6}
19: {8} 39: {2,6} 61: {18} 79: {22}
21: {2,4} 41: {13} 62: {1,11} 82: {1,13}
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Select[Range[100],PrimeOmega[#]<=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
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